There is indeed a Menger's theorem for matroids first proven by Tutte.  The reference is

>Tutte, W. T., Menger’s theorem for matroids, Journal of Research of the National
Bureau of Standards—B. Mathematics and Mathematical Physics, 69B (1965), 49–53.

This theorem is nowadays called *Tutte's linking theorem,* and it is sad that it is not more widely known.  I'll take this chance to try and popularize it.  First some notation. 

Let $M=(E,r_M)$ be a matroid and let $A$ and $B$ be disjoint subsets of $E$.
We define the *local connectivity* between $A$ and $B$ to be 

$\sqcap_M(A,B):=r_M(A)+r_M(B)-r_M(A \cup B)$.

We next define $\lambda_M(A):=\sqcap_M(A,E-A)$, and call $\lambda_M$ the *connectivity function* of $M$.  It is fairly straightforward to check that $\lambda_M$ is symmetric, submodular, invariant under duality, and monotone under taking minors.  Finally, we define 

$\kappa_M(A,B) = \min(\lambda_M(X) : A \subseteq X \subseteq E-B)$.  It is easy to show that for any $C \subseteq E - (A \cup B)$, we have $\sqcap_{M / C} (A,B) \leq \kappa_M(A,B)$.  Tutte's linking theorem says that we can always find a $C$ that gives us equality.

**Tutte's Linking Theorem.** There exists $C \subseteq E - (A \cup B)$, such that 
$\sqcap_{M / C} (A,B) = \kappa_M(A,B)$.

The proof is not very difficult, so instead I'll just briefly say why this generalizes Menger's theorem for graphs.  The form of Menger's theorem that it generalizes is

**Menger's Theorem.**  Let $a$ and $b$ be non-adjacent vertices in a graph $G$.  Let $k$ be the size of a smallest vertex cut separating $a$ and $b$.  Then there exist $k$ internally vertex disjoint paths between $a$ and $b$.

*Proof (via Tutte's Linking Theorem)*.  Let $A$ and $B$ be the sets of edges incident to $a$ and $b$ respectively.  Note that $A$ and $B$ are disjoint since $a$ and $b$ are non-adjacent.  Let $k$ be the size of the smallest vertex cut separating $a$ and $b$.  Now just apply Tutte's Linking Theorem to $A$ and $B$ together with the following two observations:

1. $k = \kappa_{M(G)}(A,B)+1,$ and
2. there exists $n$ internally vertex disjoint paths between $a$ and $b$ if and only if there exists
$C \subseteq E(G) - (A \cup B)$ such that $\sqcap_{M(G /C)}(A,B) \geq n-1$.